| L(s) = 1 | + 8·4-s − 16·7-s + 8·13-s + 42·16-s − 40·19-s + 44·25-s − 128·28-s − 56·31-s + 136·37-s + 104·43-s − 116·49-s + 64·52-s + 8·61-s + 136·64-s + 112·67-s − 448·73-s − 320·76-s − 448·79-s − 128·91-s − 152·97-s + 352·100-s − 104·103-s + 680·109-s − 672·112-s − 448·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2.28·7-s + 8/13·13-s + 21/8·16-s − 2.10·19-s + 1.75·25-s − 4.57·28-s − 1.80·31-s + 3.67·37-s + 2.41·43-s − 2.36·49-s + 1.23·52-s + 8/61·61-s + 17/8·64-s + 1.67·67-s − 6.13·73-s − 4.21·76-s − 5.67·79-s − 1.40·91-s − 1.56·97-s + 3.51·100-s − 1.00·103-s + 6.23·109-s − 6·112-s − 3.61·124-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.343997983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.343997983\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - p^{3} T^{2} + 11 p T^{4} + 3 p^{3} T^{6} - 375 T^{8} + 3 p^{7} T^{10} + 11 p^{9} T^{12} - p^{15} T^{14} + p^{16} T^{16} \) |
| 5 | \( 1 - 44 T^{2} + 2896 T^{4} - 15972 p T^{6} + 2805486 T^{8} - 15972 p^{5} T^{10} + 2896 p^{8} T^{12} - 44 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 + 8 T + 22 p T^{2} + 160 p T^{3} + 10526 T^{4} + 160 p^{3} T^{5} + 22 p^{5} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 4 T + 22 p T^{2} + 388 T^{3} + 42374 T^{4} + 388 p^{2} T^{5} + 22 p^{5} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( 1 - 1232 T^{2} + 735100 T^{4} - 289134384 T^{6} + 90323356422 T^{8} - 289134384 p^{4} T^{10} + 735100 p^{8} T^{12} - 1232 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 + 20 T + 868 T^{2} + 8916 T^{3} + 328146 T^{4} + 8916 p^{2} T^{5} + 868 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 2444 T^{2} + 3254416 T^{4} - 2852397780 T^{6} + 1774428987822 T^{8} - 2852397780 p^{4} T^{10} + 3254416 p^{8} T^{12} - 2444 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 1880 T^{2} + 1924252 T^{4} - 2074330856 T^{6} + 2128273704070 T^{8} - 2074330856 p^{4} T^{10} + 1924252 p^{8} T^{12} - 1880 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 28 T + 1432 T^{2} + 13412 T^{3} + 1149278 T^{4} + 13412 p^{2} T^{5} + 1432 p^{4} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 68 T + 4504 T^{2} - 191964 T^{3} + 7669374 T^{4} - 191964 p^{2} T^{5} + 4504 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 8624 T^{2} + 37064956 T^{4} - 103569069648 T^{6} + 204770887955334 T^{8} - 103569069648 p^{4} T^{10} + 37064956 p^{8} T^{12} - 8624 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 - 52 T + 3460 T^{2} - 22452 T^{3} + 2716146 T^{4} - 22452 p^{2} T^{5} + 3460 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 6620 T^{2} + 27373072 T^{4} - 80988531396 T^{6} + 198687369383598 T^{8} - 80988531396 p^{4} T^{10} + 27373072 p^{8} T^{12} - 6620 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 14348 T^{2} + 100839952 T^{4} - 466253289236 T^{6} + 1539595638444910 T^{8} - 466253289236 p^{4} T^{10} + 100839952 p^{8} T^{12} - 14348 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 19976 T^{2} + 192056668 T^{4} - 1159676388024 T^{6} + 4810763889113478 T^{8} - 1159676388024 p^{4} T^{10} + 192056668 p^{8} T^{12} - 19976 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 4 T + 8686 T^{2} - 150828 T^{3} + 38662374 T^{4} - 150828 p^{2} T^{5} + 8686 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 56 T + 15436 T^{2} - 560264 T^{3} + 95435590 T^{4} - 560264 p^{2} T^{5} + 15436 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 22412 T^{2} + 258098896 T^{4} - 2017610554196 T^{6} + 11687868067441198 T^{8} - 2017610554196 p^{4} T^{10} + 258098896 p^{8} T^{12} - 22412 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 224 T + 28780 T^{2} + 2677024 T^{3} + 212472614 T^{4} + 2677024 p^{2} T^{5} + 28780 p^{4} T^{6} + 224 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 224 T + 40282 T^{2} + 4437320 T^{3} + 5320162 p T^{4} + 4437320 p^{2} T^{5} + 40282 p^{4} T^{6} + 224 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 26528 T^{2} + 386566684 T^{4} - 3936191194208 T^{6} + 30559983176028934 T^{8} - 3936191194208 p^{4} T^{10} + 386566684 p^{8} T^{12} - 26528 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( 1 - 13664 T^{2} + 265530628 T^{4} - 2460680918048 T^{6} + 25256672766991750 T^{8} - 2460680918048 p^{4} T^{10} + 265530628 p^{8} T^{12} - 13664 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 + 76 T + 22576 T^{2} + 814804 T^{3} + 238699630 T^{4} + 814804 p^{2} T^{5} + 22576 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.91761390785414923583727688380, −3.75702765973370736733531876199, −3.68588437896094974569884694644, −3.63157585435489495355779529089, −3.37340254019189160780599448033, −3.22232077395230589328649847367, −3.20552549403357173148217824936, −2.89714364048054239135845832980, −2.81119449229483724614143772792, −2.74432945297414111655168425868, −2.68996916129519062626356218950, −2.67490069841919472020590064014, −2.60054413359373242370893058149, −2.17481980871747849021926304509, −1.99841489883646292178510673962, −1.76919119218187547215694575914, −1.70551406709084650467953495909, −1.65702288041194896771742927732, −1.36342041027898595886336821621, −1.18217901337544388344756200804, −1.18186370540942317942088562555, −0.64028139504803290942636210930, −0.59669814716007707397166022965, −0.29113404827464644187027136212, −0.18450046775317236672999191910,
0.18450046775317236672999191910, 0.29113404827464644187027136212, 0.59669814716007707397166022965, 0.64028139504803290942636210930, 1.18186370540942317942088562555, 1.18217901337544388344756200804, 1.36342041027898595886336821621, 1.65702288041194896771742927732, 1.70551406709084650467953495909, 1.76919119218187547215694575914, 1.99841489883646292178510673962, 2.17481980871747849021926304509, 2.60054413359373242370893058149, 2.67490069841919472020590064014, 2.68996916129519062626356218950, 2.74432945297414111655168425868, 2.81119449229483724614143772792, 2.89714364048054239135845832980, 3.20552549403357173148217824936, 3.22232077395230589328649847367, 3.37340254019189160780599448033, 3.63157585435489495355779529089, 3.68588437896094974569884694644, 3.75702765973370736733531876199, 3.91761390785414923583727688380
Plot not available for L-functions of degree greater than 10.
